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Definition. Prism- this is a polyhedron, all of whose vertices are located in two parallel planes, and in these same two planes lie two faces of the prism, which are equal polygons with, respectively parallel sides, and all edges not lying in these planes are parallel.
Two equal faces are called prism bases(ABCDE, A 1 B 1 C 1 D 1 E 1).
All other faces of the prism are called side faces(AA 1 B 1 B, BB 1 C 1 C, CC 1 D 1 D, DD 1 E 1 E, EE 1 A 1 A).
All side faces form lateral surface of the prism .
All lateral faces of the prism are parallelograms .
The edges that do not lie at the bases are called the lateral edges of the prism ( AA 1, BB 1, CC 1, DD 1, EE 1).
Prism diagonal is a segment whose ends are two vertices of a prism that do not lie on the same face (AD 1).
The length of the segment connecting the bases of the prism and perpendicular to both bases at the same time is called prism height .
Designation:ABCDE A 1 B 1 C 1 D 1 E 1. (First, in the order of traversal, the vertices of one base are indicated, and then, in the same order, the vertices of another; the ends of each side edge are designated by the same letters, only the vertices lying in one base are designated by letters without an index, and in the other - with an index)
The name of the prism is associated with the number of angles in the figure lying at its base, for example, in Figure 1 there is a pentagon at the base, so the prism is called pentagonal prism. But because such a prism has 7 faces, then it heptahedron(2 faces - the bases of the prism, 5 faces - parallelograms, - its side faces)
Among straight prisms, a particular type stands out: regular prisms.
A straight prism is called correct, if its bases are regular polygons.
Parallelepiped
Parallelepiped- This quadrangular prism, at the base of which lies a parallelogram (an inclined parallelepiped). Right parallelepiped- a parallelepiped whose lateral edges are perpendicular to the planes of the base.Rectangular parallelepiped- a right parallelepiped whose base is a rectangle.
Properties and theorems:
Some properties of a parallelepiped are similar to the known properties of a parallelogram. A rectangular parallelepiped having equal dimensions is called cube
.A cube has all equal squares.Diagonal square, equal to the sum squares of its three dimensions 
,
where d is the diagonal of the square;
a is the side of the square.
An idea of a prism is given by:
- various architectural structures;
- Kids toys;
- packaging boxes;
- designer items, etc.
The area of the total and lateral surface of the prism
Total surface area of the prism is the sum of the areas of all its faces Lateral surface area is called the sum of the areas of its lateral faces. The bases of the prism are equal polygons, then their areas are equal. That's whyS full = S side + 2S main,
Where S full- total surface area, S side-lateral surface area, S base- base area
The lateral surface area of a straight prism is equal to the product of the perimeter of the base and the height of the prism.
S side= P basic * h,
Where S side-area of the lateral surface of a straight prism,
P main - perimeter of the base of a straight prism,
h is the height of the straight prism, equal to the side edge.
Prism volume
The volume of a prism is equal to the product of the area of the base and the height.
1. Smallest number The tetrahedron has 6 edges.
2. A prism has n faces. What polygon lies at its base?
(n - 2) - square.
3. Is a prism straight if its two adjacent side faces are perpendicular to the plane of the base?
Yes it is.
4. In which prism are the lateral edges parallel to its height?
In a straight prism.
5. Is a prism regular if all its edges are equal to each other?
No, it may not be direct.
6. Can the height of one of the side faces of an inclined prism also be the height of the prism?
Yes, if this face is perpendicular to the base.
7. Is there a prism in which: a) the side edge is perpendicular to only one edge of the base; b) only one side face is perpendicular to the base?
a) yes. b) no.
8. A regular triangular prism is divided into two prisms by a plane passing through the midlines of the bases. What is the ratio of the lateral surface areas of these prisms?
By theorem 27 we find that the lateral surfaces are in the ratio 5: 3
9. Will the pyramid be regular if its side faces are regular triangles?
10. How many faces perpendicular to the plane of the base can a pyramid have?
11. Is there a quadrangular pyramid whose opposite side faces are perpendicular to the base?
No, otherwise there would be at least two straight lines passing through the top of the pyramid, perpendicular to the bases.
12. Can all the faces of a triangular pyramid be right triangles?
Yes (Figure 183).
General information about straight prism
The lateral surface of a prism (more precisely, the lateral surface area) is called sum areas of the side faces. Full surface prism is equal to the sum of the lateral surface and the areas of the bases.
Theorem 19.1. The lateral surface of a straight prism is equal to the product of the perimeter of the base and the height of the prism, i.e., the length of the side edge.
Proof. The lateral faces of a straight prism are rectangles. The bases of these rectangles are the sides of the polygon lying at the base of the prism, and the heights are equal to the length of the side edges. It follows that side surface prism is equal
S = a 1 l + a 2 l + ... + a n l = pl,
where a 1 and n are the lengths of the base edges, p is the perimeter of the base of the prism, and I is the length of the side edges. The theorem has been proven.
Practical task
Problem (22) . In an inclined prism it is carried out section, perpendicular to the side ribs and intersecting all the side ribs. Find the lateral surface of the prism if the cross-sectional perimeter is equal to p and the side edges are equal to l.
Solution. The plane of the drawn section divides the prism into two parts (Fig. 411). Let us subject one of them to parallel translation, combining the bases of the prism. In this case, we obtain a straight prism, the base of which is the cross-section of the original prism, and the side edges are equal to l. This prism has the same lateral surface as the original one. Thus, the lateral surface of the original prism is equal to pl.
Summary of the covered topic
Now let’s try to summarize the topic we covered about prisms and remember what properties a prism has.
Prism properties
Firstly, a prism has all its bases as equal polygons;
Secondly, in a prism all its lateral faces are parallelograms;
Thirdly, in such a multifaceted figure as a prism, all lateral edges are equal;
Also, it should be remembered that polyhedra such as prisms can be straight or inclined.
Which prism is called a straight prism?
If the prism side rib is located perpendicular to the plane of its base, then such a prism is called a straight line.
It would not be superfluous to recall that the lateral faces of a straight prism are rectangles.
What type of prism is called oblique?
But if the side edge of a prism is not located perpendicular to the plane of its base, then we can safely say that it is an inclined prism.
Which prism is called correct?
If a regular polygon lies at the base of a straight prism, then such a prism is regular.
Now let us remember the properties that a regular prism has.
Properties of a regular prism
Firstly, regular polygons always serve as the bases of a regular prism;
Secondly, if we consider the side faces of a regular prism, they are always equal rectangles;
Thirdly, if you compare the sizes of the side ribs, then in a regular prism they are always equal.
Fourthly, a correct prism is always straight;
Fifthly, if in a regular prism the lateral faces have the shape of squares, then such a figure is usually called a semi-regular polygon.
Prism cross section
Now let's look at the cross section of the prism:
Homework
Now let's try to consolidate the topic we've learned by solving problems.
Let's draw an inclined triangular prism, the distance between its edges will be equal to: 3 cm, 4 cm and 5 cm, and the lateral surface of this prism will be equal to 60 cm2. Having these parameters, find the side edge of this prism.
Do you know that geometric figures constantly surround us not only in geometry lessons, but also in Everyday life There are objects that resemble one or another geometric figure.
Every home, school or work has a computer whose system unit is shaped like a straight prism.
If you pick up a simple pencil, you will see that the main part of the pencil is a prism.
Walking along main street city, we see that under our feet lies a tile that has the shape of a hexagonal prism.
A. V. Pogorelov, Geometry for grades 7-11, Textbook for educational institutions